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Chromatic homotopy theory of spaces

Project description

Study sheds more light on the chromatic decomposition of spaces

Chromatic homotopy theory – a subfield of stable homotopy theory – studies complex-oriented cohomology theories from the ‘chromatic’ point of view. This method decomposes a spectrum into monochromatic pieces. Each piece is a localised structure corresponding to one of the prime fields of higher algebra. The goal of the EU-funded ChromSpaces project is to study the chromatic decomposition of spaces rather than of a spectrum. It will establish structural results for the category of all monochromatic spaces ‘of a given colour’ and investigate how the original space can be built from its local pieces. The new techniques will be based on previous results associating monochromatic spaces to spectral Lie algebras, which generalise Quillen’s rational homotopy theory to all the other relevant chromatic localisations of homotopy theory.

Objective

Many current developments in stable homotopy theory are guided by the ‘chromatic perspective’. One decomposes a spectrum into its monochromatic pieces, each of which is a localization corresponding to one of the prime fields of higher algebra (the Morava K-theories, generalizing the prime fields Q and F_p of ordinary algebra). The goal of this proposal is to study the chromatic decomposition of spaces, as opposed to that of spectra. I will establish structural results for the category of all monochromatic spaces ‘of a given color’ and study the assembly question: how to put the pieces back together to retrieve information about the original space? The techniques are informed by my recent results relating monochromatic spaces to spectral Lie algebras, which generalize Quillen’s rational homotopy theory to all the other relevant chromatic localizations of homotopy theory. More precisely, this research has the following goals. 1. Develop the structure theory of spectral Lie algebras and apply it to monochromatic spaces. This includes understanding Koszul duality between spectral Lie algebras and commutative ring spectra, with applications to a conjecture of Francis-Gaitsgory, and decomposition results for spectral Lie algebras, with applications to torsion exponents of homotopy groups, building on classical work of Cohen-Moore-Neisendorfer. 2. Develop a theory of transchromatic spectral Lie algebras, explaining how the different monochromatic pieces of homotopy theory interact. This connects to my previous work on the Goodwillie tower of homotopy theory and Tate coalgebras.

Host institution

UNIVERSITEIT UTRECHT
Net EU contribution
€ 1 500 000,00
Address
HEIDELBERGLAAN 8
3584 CS Utrecht
Netherlands

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Region
West-Nederland Utrecht Utrecht
Activity type
Higher or Secondary Education Establishments
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Total cost
€ 1 500 000,00

Beneficiaries (1)